A New Approach for Determining Phase Response Curves Reveals that Purkinje Cells Can Act as Perfect Integrators

Abstract

Cerebellar Purkinje cells display complex intrinsic dynamics. They fire spontaneously, exhibit bistability, and via mutual network interactions are involved in the generation of high frequency oscillations and travelling waves of activity. To probe the dynamical properties of Purkinje cells we measured their phase response curves (PRCs). PRCs quantify the change in spike phase caused by a stimulus as a function of its temporal position within the interspike interval, and are widely used to predict neuronal responses to more complex stimulus patterns. Significant variability in the interspike interval during spontaneous firing can lead to PRCs with a low signal-to-noise ratio, requiring averaging over thousands of trials. We show using electrophysiological experiments and simulations that the PRC calculated in the traditional way by sampling the interspike interval with brief current pulses is biased. We introduce a corrected approach for calculating PRCs which eliminates this bias. Using our new approach, we show that Purkinje cell PRCs change qualitatively depending on the firing frequency of the cell. At high firing rates, Purkinje cells exhibit single-peaked, or monophasic PRCs. Surprisingly, at low firing rates, Purkinje cell PRCs are largely independent of phase, resembling PRCs of ideal non-leaky integrate-and-fire neurons. These results indicate that Purkinje cells can act as perfect integrators at low firing rates, and that the integration mode of Purkinje cells depends on their firing rate.

In vivo cardiac phase response curve elucidates human respiratory heart rate variability

“…A key feature determining the interaction properties is the ability of an oscillator to respond, by shifting its phase, to an external perturbation; this feature is often quantified in terms of phase response curves (PRC), which describe both reactions to a single pulse perturbation and to a continuous force, resulting in a continuous phase shift in the latter case 2,3,7,8 . This approach is used in neuroscience [9][10][11] , cardiorespiratory physiology [12][13][14][15][16][17] and chronobiology 18,19 , to name just a few. A traditional experimental approach to obtain the PRC implies that the oscillator (for example, a neuron) is isolated from the environment (for example, from other neurons that normally interact with it) and is repeatedly perturbed by (weak) short pulses 20 .…”

Efficient estimation of phase-response curves via compressive sensing

“…In such cases, the exact BP algorithm, a priori assuming that the recovered signal predicts the measurement results exactly, may fail to converge. Furthermore, the Dantzig selector also applies to cases where the signal is approximately sparse such that many coefficients actually remain nonzero, but their magnitudes decay at least faster than a power law, |⌬ k | ϳ 1⁄k n for a positive integer n. This is important since a PRC can have this property: for example, it has been reported that cerebellar Purkinje neuron can have a square wave-like PRC, indicating neuronal near-perfect integration (Phoka et al 2010), and in this case many Fourier coefficients will be nonzero but decay as |⌬ k | ϳ 1⁄k.…”

“…We could not observe similar bias in our simulated or experimental data and did not try to prevent it. However, in principle, we can address this problem in the continuous stimulus paradigm in a similar way as Phoka et al (2010) by using corrections from multiple ISIs: we first make the stimulus-IFRC pair from every two consecutive ISI, such as {T i ϩ T i ϩ 1 } and {[x i , x i ϩ 1 ]}, and estimate the PRC for these two ISIs as [⌬;⌬]. In this way, we can eliminate the effect of the spike time uncertainty in the spike between the i-th and (i ϩ 1)-th ISI.…”

“…At the level of a single neuron, the PRC characterizes the intrinsic properties of the neuron and determines how its spiking pattern will change with input. For example, whether the PRC is monophasic or biphasic indicates whether the neuron is an integrator or resonator based on its excitability class (Ermentrout 1996;Gutkin et al 2005;Hansel et al 1995;Izhikevich 2010;Moehlis et al 2006;Phoka et al 2010;Stiefel et al 2008). Beyond the single cell level, the PRC has been useful for predicting population behavior such as synchrony.…”

“…Therefore, a priori, the degrees of freedom, or dimensionality, of the data (probe stimuli) are much larger than those actually constituting the PRC, and this becomes one of the main challenges in PRC estimation. Additional problems arise in practice when the firing is not perfectly regular and the period has to be estimated, rather than directly measured, together with the phase shifts (Phoka et al 2010).…”